LIBRARY 


UNIVERSITY  OF  CALIFORNIA. 


Class 


Copyright,  1899,  1905,  by 
CHARLES  H.  SWAN  and  THEODORE  HORTON. 


HYDRAULIC   DIAGRAMS 


FOR  THE 


DISCHARGE  OF  CONDUITS 
AND  CANALS 

Based  upon  the  Formula  of  Gangfuillet  and  Kutter 


BY 

CHARLES  H.  SWAN,  M.  Am.  Soc.  C.  E., 
p 

AND 

THEODORE  HORTON,  M.  Am.  Soc,  C.  E., 

WITH 

A  Description  of  the  Diagrams  and  their  Use  by  Theodore  Horton 


SECOND  EDITION 


NEW  YORK  : 

THE  ENGINEERING  NEWS  PUBLISHING  COMPANY 
1905 


< 


PREFACE    TO SECOND    EDITION. 


The  following  set  of  diagrams,  based  upon  the  formula  of  Gan- 
guillet  and  Kutter,  is  intended  for  use  in  the  study  of  such  sections 
of  conduits  and  canals  as  are  commonly  employed  in  sewerage, 
water  supply,  water  power  and  land  drainage.  Tlje  set  includes 
conduits  of  eight  different  types  of  cross-section,  and  canals  of 
rectangular  and  trapezoidal  cross-section. 

In  presenting  this  set  of  diagrams,  it  has  been  the  aim  of  the 
authors  to  cover  the  field  with  as  limited  a  number  of  diagrams 
as  will  readily  conform  to  a  simple  and  practical  system  for  use. 
A  short  discussion  of  the  formula  and  a  description  of  the  dia- 
grams and  their  use  appear  on  the  following  pages. 

In  this  edition  one  diagram  has  been  added  to  the  set.  This 
diagram,  No.  I,  gives  discharges  and  velocities  for  the  smaller 
diameters  of  conduits  on  a  scale  much  larger  than  was  used  on 
diagram  No.  I  of  the  previous  edition.  No  changes  have  been 
made  in  the  text  other  than  italicizing  certain  phrases  for  the 
purpose  of  emphasis. 


CONTENTS. 


Chapter  Page 

I.     The  Formula  of  Ganguillet  and  Kutter 5 

II.     Description  of  the  Diagrams   8 

Conduits 8 

Canals 1 1 

III.     Use  of  the  Diagrams 12 

Conduits :    Flowing  Full   13 

Flowing  Partially  Full   15 

Under  Pressure   16 

Canals :    Class  A.     Sections  in  which  the   Ratio  of 

Depth  of  Flow  to  Depth  of  Filled  Section  is  i.oo. .  17 
Class  B.    Sections  in  which  the  Ratio  of  Depth  of 
Flow  to  Depth  of  Filled  Section  is  Greater  or  Less 

than   i.oo   .  18 


DIAGRAMS. 

(Following  text.) 

DISCHARGE  FROM   CIRCULAR  CONDUITS   FLOWING 
FULL  WITH  N  =  =  0.015., 

1.  Diameters  of  3  ins.  to  I  ft.  6  ins.  and  Hydraulic  In- 

clinations of  o.io  to  o.oooi. 

2.  Diameters  of  4  ins.  to  4  ft.  and  Hydraulic  Inclinations 

of  o.io  to  o.oooi. 

3.  Diameters  of  6  ins.  to  10  ft.  and  Hydraulic  Inclinations 

of  0.006  to  0.000025. 

4.  Diameters  of  4-J  to  20  ft.  and  Hydraulic  Inclinations 

of  0.0055  to  0.000025. 

5.     Ratios  of  Discharge  for  Different  Values  of  n  to  Discharge 
for  n  =  0.015,  for  Circular  Sections. 

RATIOS    OF    HYDRAULIC    ELEMENTS    OF    VARIOUS 

SECTIONS. 

6.  Circular  Section. 

7.  Gothic  Section. 

8.  Basket  Handle  Section. 

9.  Catenary  Section. 

10.  Egg-Shaped  Section. 

11.  Square  Section. 

12.  Horseshoe  Section  (Wachusett  Aqueduct). 

13.  Horseshoe  Section  (Croton  Aqueduct). 

14.  Discharge  from  Rectangular  Filled  Sections,  n  =  0.025. 

15.  Ratios  of  Discharge  for  Different  Values  of  n  to  Discharge 

for  n  =  0.025,  for  Rectangular  and  Trapezoidal  Sections. 

16.  Ratio  of  Discharge  of  Filled  Segment  to  that  of  Filled  Sec- 

tion. 

17.  Ratio  of  Area  of  Segment  to  Area  of  Filled  Section. 


HYDRAULIC    DIAGRAMS    FOR    THE    DISCHARGE    OF 
CONDUITS  AND  CANALS. 


CHAPTER   I. 


The  Formula  of  Ganguillet  and  Kiuter. 

This  formula  for  the  mean  velocity  of  discharge  of  rivers,  canals 
and  conduits  was  obtained  from  a  comparison  of  numerous  ex- 
pe'riments  made  in  different  countries  upon  natural  and  artificial 
water  courses  of  many  sizes  and  various  kinds  of  materials. 

The  formula  assumes  that  a  uniform  flow  has  been  established 
and  gives  the  equation  of  the  mean  velocity  of  flow.  This  equa- 
tion is  as  follows  for  metric  measures : 


v=H 


(23  + 


0.00155 


VRS 


When  reduced  to  measures  in  English  feet  it  becomes 

1.81132      0.00281 

41. 6603  H \ 

n  S 


v= 


0.00281  \     n 

i  +  (41.6603  + )-^r- 

s      /  J  T? 


It  may  he  expressed  more  briefly, 


in  which 


V=  the  mean  velocity  of  flow  ; 

c  =  the  velocity  coefficient  ; 

R=  the  mean  hydraulic  radius  of  the  stream  ; 


\ 

6  HYDRAULIC   DIAGRAMS. 

S  =  the  sine  of  the  inclination,  or  fall  in  a  unit  of  length ; 

n  =  a  frictional  factor  dependent  upon  the  nature  of  the  surface 
over  which  the  water  flows. 

For  brevity,  let  us  substitute  letters  for  the  numbers  in  the  for- 
mula. We  mav  then  write 


1 

m 

n 

S 

v  —  < 

V         1 

m 

T  -\-(x  4- 

"I    n 

i    i  \a  ~r 

S 

JTS 

m\  1  m 
and  by  substituting  x  =  (a  H I  n  and  z  =  a  -f-  —  H we  may 

S/  n  S 

write 


V  RS 

i  -i 

VRJ 

in  which 


c  = 


i  + 


When  the  quantity  V  is  sought  from  general  data,  and  the  co- 
efficient c  is  not  needed  separately,  the  following  transformation 
may  be  made  : 


which  is  a  useful  form  from  which  tables  may  be  calculated,  and,  as 
was  done  with  many  of  the  present  diagrams,  the  results  plotted. 
Since  a  proper  selection  of  the  friction  factor  n  is  essential  in 
the  application  of  the  above  general  equation  and  the  use  of  the 
present  diagrams  based  thereon,  the  following  values  for  it  will 
be  here  reproduced  for  a  general  guide  in  practice  : 


HYDRAULIC  DIAGRAMS.  7 

n  =  .007  to  .008 :  Glass,  new  tin,  lead  and  galvanized  iron  pipe, 
n  —  .008  to  .009:  New    seamless    wrought-iron    and    new-coated 
cast-iron  pipe  in  best  of  condition  and  aline- 
ment. 

n  =  .009  to  .010 :  New  cast-iron  pipe,  new  enamelled  and  glazed 
pipe  of  all  sorts ;  well  planed  timber  in  per- 
fect alinement. 

n  =  .010  to  .on:  New  wrought-iron  riveted  pipe  of  small  diam- 
eter ;  new  wooden  stave  pipe ;  planed  tim- 
ber, neat  cement. 

n  —  .on  to  .012:  Unplaned  timber  carefully  joined;  cement,  one- 
third  sand ;  new  terra  cotta ;  new  well-laid 
brickwork,  carefully  pointed  and  scraped ; 
clean  cast-iron  pipe  in  use  some  time. 

n  =  .012  to  .013  :  Unplaned  timber  ;  cement  two-thirds  sand  ; 
ashlar  and  well-laid  brickwork;  ordinary 
brickwork  plastered ;  earthen  and  stone- 
ware pipes  in  good  condition  but  not  new; 
plaster  and  planed  wood  of  inferior  qual- 
ity ;  glazed  pipe  poorly  laid  or  foul  from 
use ;  new  wrought-iron  riveted  pipe  with 
many  joints  and  rivets. 

n  =  .015:  Rough-faced  brickwork;  ashlar  and  well-laid  brick- 
work slightly  deteriorated  from  use ;  fouled  or 
slightly  tuberculated  cast-iron  pipe ;  large  wrought- 
iron  riveted  pipe,  few  years  in  use  but  in  good  con- 
dition; canvas  lining. 

n  =  .017 :  Brickwork  and  ashlar  in  inferior  condition  or  badly 
fouled ;  tuberculated  and  fouled  iron  pipe ;  rubble 
in  cement  or  plaster,  in  good  condition;  gravel- 
lined  canals  with  f-in.  grains  well  rammed  or  ce- 
ment grouted. 

n  =  .020:  Rubble  in  cement  of  inferior  quality;  coarse  rubble 
set  dry ;  brickwork  in  bad  condition ;  gravel-lined 
canals,  with  one-inch  grains,  well  rammed  or  ce- 
ment grouted. 

n  =  .0225 :  Rough  rubble  in  bad  condition ;  canals  with  earthen 
beds  in  perfect  order  and  alinement. 


8  HYDRAULIC   DIAGRAMS. 

n  =  .025 :  Canals  with  earthen  beds  in  good  order  and  alinement 

and  free  from  stones  and  weeds, 
n  =  .030:  Canals  with  earthen  beds  in  moderately  good  order 

and  alinement,  with  few  stones  and  weeds. 

n  =  .040:  Canals  with  earthen  beds  in  bad  condition  and  aline- 
ment, having  stones  and  weeds  in  great  quantity. 
For  a  more  complete  statement  of  the  formula  and  its  derivation 
the  reader  is  referred  to  the  following  well-known  books: 

"The  New  Formula  for  Mean  Velocity  of  Discharge  of  Rivers 
and  Canals."  By  W,  R.  Kutter.  Translated  from  Articles 
in  the  Cultur-Ingenieur  by  Louis  D'A.  Jackson.  London, 
1876. 

"Flow  of  Water  in  Rivers  and  other  Channels."  By  Ganguillet 
and  Kutter.  Translated,  with  numerous  additions,  by  R. 
Hering  and  J.  C.  Trautwine.  New  York,  1891." 


CHAPTER    II. 


Description  of  the  Diagrams. 

For  convenience  in  their  description,  the  diagrams  will  be  con- 
sidered under  the  following  two  groups :  The  first  group,  Diagrams 
I  to  13,  inclusive,  deal  with  circular  and  similar  types  of  con- 
duits, more  especially  applicable  to  the  conveyance  of  moderate 
volumes  of  water  and  sewage,  flowing  either  openly  or  under  pres- 
sure. The  second  group,  Diagrams  14  to  17,  inclusive,  deal  with 
canals  of  rectangular  and  trapezoidal  cross  sections,  more  ap- 
plicable to  the  conveyance  of  larger  volumes  of  water  and  sewage, 
not  flowing  under  pressure. 

Conduits. 

Diagrams  I — 4:  Of  the  first  group,  Diagrams  I,  2,  3  and  4  are 
discharge  diagrams,  very  similar  in  construction,  and,  differing 
only  in  range  of  data,  will  be  described  together.  They  give  veloci- 
ties in  feet  per  second,  and  discharges  in  cubic  feet  per  second, 
from  circular  conduits,  running  full,  for  diameters  ranging  from  3 
ins.  to  20  ft.;  hydraulic  inclinations  from  0.000025  to  o.io;  with  a 
friction  factor  n  — .015. 

The  vertical  scale  on  each  diagram  represents  hydraulic  inclina- 


HYDRAULIC  DIAGRAMS.  9 

tions,  expressed  fractionally  at  the  right  and  trigonometrically  (i.  e. 
the  sine  of  the  angle  of  inclination)  at  the  left.  Also,  at  the  left  is 
given  a  scale  of  corresponding  square  roots  of  the  slope.  The  hori- 
zontal scale  represents  discharges  in  cubic  feet  per  second.  Begin- 
ning at  the  left  on  Diagrams  3  and  4  this  scale  is  broken  at  inter- 
vals, and  at  the  same  time  increased  in  value,  thus  giving  the 
diagrams  a  wider  range  of  data  than  could  be  obtained  by  the  use 
of  a  single  scale.  The  radial  lines  represent  diameters  in  feet  and 
fractions  thereof.  By  selecting  a  natural  vertical  scale,  based  upon 
the  square  root  of  the  slope,  these  lines  become  straight  between 
successive  divisions  of  the  horizontal  scale,  a  feature  which  not 
only  facilitated  the  construction  of  the  diagrams  but  allows  them  to 
be  readily  extended  beyond  their  present  limits  in  any  special  case. 

Diagram  5 :  As  already  stated,  the  discharge  diagrams  are  based 
upon  a  friction  factor  n  =  .oi5.  Diagram  5,  consisting  of  four 
curves,  representing  different  diameters,  shows  the  relation  be- 
tween the  discharge  for  n  — .015  and  discharges  for  other  values 
of  n,  ranging  from  .008  to  .018.  For  simplicity,  this  diagram  may 
be  considered  a  correction  diagram  on  which  the  vertical  scale 
represents  friction  factors  and  the  horizontal  scale  correction  co- 
efficients to  be  applied  to  discharges  for  n  —  .015.  The  four  curves 
intersect  at  a  common  point  whose  abscissa  is  i.oo  and  ordinate 
.015,  as  might  be  expected.  For  diameters  intermediate  to  those 
represented  by  the  four  curves  interpolation  will,  of  course,  be 
necessary. 

Diagrams  6 — 13  :  The  remaining  eight  diagrams  of  the  first  group, 
termed  for  convenience  ratio  diagrams,  give  for  each  of  the  types 
of  conduits  considered  the  ratio  of  each  of  the  three  elements — • 
area,  mean  velocity  and  discharge — of  the  "filled  segment"  to  that 
of  the  "filled  section"  corresponding  to  any  ratio  of  depth  of  flow 
to  the  vertical  diameter.  "Filled  segment"  refers  to  the  cross  sec- 
tion of  the  stream  of  the  partially  filled  conduit,  and  "filled  section" 
to  the  cross  section  of  the  entire  conduit.  The  vertical  scale 
represents  the  ratio  of  the  depth  of  flow  to  the  vertical  diameter 
and  the  horizontal  scale  corresponding  ratios  of  the  hydraulic 
elements  of  the  filled  segment  to  those  of  the  filled  section. 

Expressed  also  on  each  diagram  are  other  hydraulic  elements  of 
the  filled  section  in  terms  of  the  vertical  or  horizontal  diameters ;  re- 


10  HYDRAULIC   DIAGRAMS. 

lations  between  the  various  geometrical  elements  for  the  purpose  of 
outlining:  the  section  ;  and  actual  data  from  which  the  curves  were 
constructed.  These  curves  are  strictly  correct  only  for  the  data  given, 
but  they  vary  so  slightly  for  different  sizes,  grades  and  friction  factors 
that  they  may  be  considered  practically  independent  of  them. 

The  symbols  and  terms  used  on  all  ratio  diagrams  are  those  com- 
monly employed  in  hydraulic  work  of  this  nature ;  the  perimeter,  P, 
the  hydraulic  radius,  R,  and  area,  A,  referring:,  however,  to  the  cross 
section  of  the  entire  conduit.  The  term  "  Equivalent"  symbolized 
thus,  =O  =  ,  when  applied  to  the  circle  means  one  of  equal  carrying 
capacity  and  not  of  equal  area. 

In  general  these  ratio  diagrams  represent  types  of  sections  com- 
monly employed  in  practice,  the  different  shapes  of  cross  sections 
possessing  either  structural  or  hydraulic  advantages.  The  circular 
section  (Diag.  6)  is  the  one  most  commonly  employed  in  practice, 
combining  strength  with  simplicity  and. economy  of  construction. 
The  Gothic  section  (Diag.  7)  combines  the  advantages  of  the  circu- 
lar section  with  increased  strength  of  the  Gothic  arch.  The  catenary 
section  (Diag.  9)  is,  theoretically,  the  section  of  greatest  strength, 
but  has  the  disadvantage  of  relatively  low  velocity  for  low  flow. 
The  egg-shaped  section  (Diag.  10),  used  somewhat  extensively  on 
combined  systems  of  sewerage,  has  the  hydraulic  advantage  of 
relatively  high  velocity  for  low  flow.  The  horse  shoe  sections 
(Diagrams  12  and  13),  used  extensively  for  large-sized  conduits, 
possess  great  stability  with  the  additional  advantage  of  economy 
in  material  and  trench  excavation.  They  have,  however,  the  dis- 
advantage of  relatively  low  velocity  for  low  flow.  The  basket 
handle  section  (Diag.  8)  is  a  modification  of  the  horse  shoe  section 
in  which  the  Gothic  arch  and  rounded  corners  were  intended  to 
give  greater  stability.  The  square  section  (Diag.  u)  is  not  com- 
mon, but  is  occasionally  used  for  wooden  sewers  and  in  other 
special  cases ;  a  little  study  will  show  its  applicability  to  any  rec- 
tangular section  of  moderate  size  which  does  not  flow  full. 

The  Gothic,  basket  handle  and  catenary  sections  have  been  used 
extensively  on  the  Metropolitan  sewers  in  Massachusetts ;  the  horse 
shoe  section  (Diag.  12)  on  the  Wachusett  aqueduct,  and  the  horse 
shoe  section  (Diag.  13)  on  the  Croton  aqueduct.  The  other  sections 
are  used  more  or  less  extensively  in  these  and  other  localities. 


HYDRAULIC  DIAGRAMS.  II 

Canals. 

The  four  diagrams  of  the  second  group,  though  referring  to 
water  courses  of  a  different  hydraulic  nature,  are  constructed  upon 
lines  closely  analagous  to  those  of  the  first  group  and  will  be  com- 
pared closely  with  them. 

With  conduits,  which  are  essentially  closed  channels,  we  have, 
for  a  full  section  of  any  type,  a  fixed  relation  between  the  various 
geometrical  elements  of  the  cross  section,  while  with  canals,  in 
order  to  secure  this  fixed  geometrical  relation,  we  must  assume  a 
definite  ratio  between  some  of  the  elements.  The  length  of  base 
and  depth  of  flow  are  chosen  in  the  present  instance,  and  a  full 
section  will  here  be  assumed  as  one  whose  depth  of  flow  is  equal 
to  one-half  the  base.  That  is,  if  a  semicircle  be  described  upon 
the  base  of  a  rectangular  or  trapezoidal  section,  a  line  drawn 
tangent  to  this  semicircle  and  parallel  with  the  base,  will  represent 
the  flow  line  of  a  filled  section.  The  ratio  assumed  is  one  which 
combines  simplicity  with  economy  of  section,  it  being  the  theoreti- 
cally economical  ratio  for  rectangular  sections  and  is  the  ratio  from 
which  no  great  variation  might  be  expected  in  actual  practice. 

For  similar  sections,  then,  both  the  geometrical  and  hydraulic 
elements  of  filled  sections  become  functions  of  the  base,  and  conse- 
quently discharge  and  ratio  diagrams  may  be  constructed  in  which 
these  elements  are  given  in  terms  of  the  length  of  base,  and  the 
depth  of  filled  section. 

Diagram  14:  This  diagram  gives,  for  rectangular  filled  sections, 
discharges  in  cubic  feet  per  second,  with  corresponding  velocities 
in  feet  per  second,  for  lengths  of  base  varying  from  10  to  50  ft., 
hydraulic  inclinations  from  0.000004  to  o.ooi,  and  with  a  friction 
factor  n  =  .025.  The  construction  and  appearance  of  this  diagram 
is  so  similar  to  the  four  discharge  diagrams  for  circular  conduits 
that  further  description  seems  unnecessary. 

Diagram  15  :  This  diagram  gives  the  relation  between  discharges 
from  actual  or  equivalent  rectangular  filled  section  for  n  =  .025  and 
discharges  with  other  values  of  n  ranging  from  .015  to  .040.  The 
diagram  consists  of  three  curves,  and  is  similar  in  every  way  to 
Diagram  4  for  conduits. 

Diagrams  16  and  17 :  These  two  diagrams,  termed  ratio  diagrams, 
are  analagous  to  the  eight  ratio  diagrams  for  conduits,  and  give  for 


12  HYDRAULIC   DIAGRAMS. 

canals  the  ratio  of  the  two  hydraulic  elements — area  and  discharge 
— of  the  filled  segment  to  that  of  the  filled  section  corresponding 
to  any  ratio  of  depth  of  flow  to  depth  of  the  filled  section.  These 
ratio  diagrams  differ  from  the  previous  ratio  diagrams  in  that  one 
of  them,  Diagram  16,  refers  exclusively  to  discharges,  and  the 
other,  Diagram  17,  exclusively  to  areas. 

Since  the  ratio  of  the  depth  of  flow  to  the  length  of  base  of  a 
filled  section  is  merely  an  assumed  one,  and,  in  reality,  the  section 
may  flow  at  a  relatively  greater  depth,  the  vertical  scales  on  these 
two  ratio  diagrams  are  extended  above  the  ratio  i.oo  sufficiently 
to  cover  all  cases  which  would  probably  arise  in  practice. 

The  table  on  Diagram  16  gives  for  trapezoidal  filled  sections, 
having  various  side  slopes,  the  equivalent  bases  of  rectangular 
filled  sections  of  equal  carrying  capacity.  A  similar  table  on 
Diagram  17  gives  simple  equations  for  obtaining  the  area  of 
trapezoidal  filled  sections  directly  in  terms  of  the  length  of  base. 
The  mean  velocity  in  all  cases  is  obtained  by  dividing  the  discharge 
by  the  area. 


CHAPTER    III. 


Use  of  the  Diagrams. 

In  this  chapter  the  following:  notation  will  be  observed  : 

Qf,  Vf,  Af,  and  Df  =  Discharge,  mean  velocity,  area  and  ver- 
tical diameter  (or  depth)  of  the  full  sec- 
tion: 

Q3,  Vs,  As,  and  Ds  =  Discharge,  mean  velocity,  area  and  depth 

of  the  filled  segment. 

Q8  V8  A8  D8 

— , — , — , — ,=  Ratios  of   the  hydraulic   elements  of   the  filled 

Qf  Vf  Af  Df  segment  to  those  of  the  filled  section,  ex- 

pressed decimally. 

H  =  horizontal  diameter  of  conduit. 

B  —  base  of  canal  section. 

Slopes  =  side  slopes  of  trapezoidal  canal  section. 

s  =  hydraulic  slope  or  sine  of  angle  of  inclination. 

n  =  friction  factor  dependent  upon  character  of  internal  surface. 


HYDRAULIC  DIAGRAMS  13 

Conduits. 

Conduits  may  flow  full,  partially  full  or  under  pressure.  They 
are  usually  designed  to  carry  a  uniform  flow  under  the  varying 
conditions  of  Q  (or  V),  D,  s  and  n.  Three  of  these  conditions 
are  usually  given  or  assumed,  and  the  remaining  one  may  be  ob- 
tained by  the  use  of  the  diagrams.  The  following  rules,  classified 
under  the  three  conditions  of  flow,  will  illustrate  the  proper  method 
of  using  the  diagrams  and  will  serve  as  a  special  guide  in  practice. 

Class  A. — Conduits  flowing  full. 

When  the  section  is  circular  the  following  principal  cases  (1-4) 
may  occur: 

1 i )  Given  Df,  s  and  n,  to  obtain  Qf.     With  Df  and  s  find  Qf  for  n 
=  .015  from  Diagram  i,  2,  3  or  4.     The  product  of  this  Qf  and  the 
ratio  on  Diagram  5  corresponding  to  given   value  of  n  will  give 
required  Qf. 

(2)  Given  Qf,  s  and  n  to  obtain  D  :  Divide  Qf  by  ratio  on  Diagam 
5  corresponding  to  given  n.      With  this   Qf  corresponding  to  n 
=  .015,  and  given  s,  find  required  Df  from  Diagrams  i,  2,  3  or  4. 
This  case  is  slightly  tentative,  since  Df  is  required  in  the  use  of 
Diagram  5.     It  will  be  seen,  however,  that  ratios  on  diagram  5 
corresponding  to  any  friction  factor,  n,  vary  but  slightly  for  a  wide 
range  in  values  of  D,  so  that  a  value  of  D  sufficiently  accurate  to 
use  Diagram  5  may  be  first  made  by  inspection 

(3)  Given  Qf,  Df  and  n  to  obtain  s :  Divide  given  Qf  by  ratio  on 
Diagram  5  corresponding  to  given  n.     With  this  Qf  corresponding 
to  n  =  .015,  and  given  Df,  find  required  s  from  Diagrams  I,  2,  3  or  4. 

(4)  Given  Qf  (obtained  from  direct  observation),  Df  and  s  to 
obtain  n :  With  given  Df  and  s  find  Qf  from  Diagrams  i,  2,  3  or  4 
for  n  —  .015.     Divide  observed  Qf  by  Qf  for  n  =  .015,  and  with 
this  ratio  find  required  n  from  Diagram  5. 

In  the  above  four  cases  Vf  may  be  substituted  for  Qf,  since  Dia- 
grams i,  2,  3  and  4  give  Vf  corresponding  to  Qf  and  since  for  any 
diameter,  Vf  varies  directly  as  Qf.  Af  in  these/ cases  is  obtained 
geometrically  from  Df. 

In  the  next  four  cases,  in  which  the  section  is  not  circular,  as  for 
instance  the  Gothic,  egg-shaped  and  horse  shoe,  a  comparison  is 


UNIVERSITY 

OF 


I4  HYDRAULIC   DIAGRAMS. 

necessary  between  the  vertical  diameters  of  these  sections,  and  the 
diameters  of  circular  sections  of  equal  carrying  capacity. 

The  equations  for  this  comparison  are  given  on  each  of  the  ratio 
diagrams,  as  are  also  the  geometrical  relations,  in  terms  of  the  ver- 
tical or  horizontal  diameter,  necessary  to  outline  the  section.  The 
Gothic  section  will  be  chosen  as  an  example  of  these  different 
types  of  sections  and  the  rules  given  above  when  applied  to  this 
section  become  as  follows : 

(5)  Given  Df,  s  and  n  of  a  Gothic  section  to  obtain  Qt :  From 

Df 

Diagram  7,   Diam.   of  —  C^=  circle  =  .     With  this    Diam. 

1.1056 

and  given  s  find  Qf  for  n  =  .015  from  Diagrams  i,  2,  3  or  4.  The 
product  of  this  Qf  and  the  ratio  on  Diagram  5  corresponding  to 
given  n,  will  give  required  Qf. 

(6)  Given  Qf,  s  and  n  of  a  Gothic  section  to  obtain  Df  and  out- 
line the  section.     With  given  Qf,  s  and  n  obtain  Df  of  a  circular 
section  by  (A-2).     With  this  Df  obtain,  from  Diagram  7,  required 
Df  of  Gothic  section  by  equation  Vert.  Diam:  =  1.1056  X  Diam.  of 
=  O—  circle.     With  Df  thus  found  outline  section  from  relation  of 
geometrical  elements  on  Diag.  7. 

(7)  Given  Df,  n  and  Qf  of  a   Gothic  section  to  obtain  s.     By 

Df 

Diagram  7,  Diam.  of  =O=  circle  =  -     — .     Divide  given  Qf  by 

1.1056 

ratio  on  Diagram  5,  corresponding  to  giveu  n.  With  this  Q  t 
corresponding  to  n  =  .015,  and  Diam.  of  =O=  circle,  obtain 
required  s  from  diagrams  i,  2,  3  or  4. 

(8)  Given  Df,  s  and  Qf  (obtained  from  direct  observation)  of  a 
Gothic  section  to  obtain  n.    From  Diag.  7,  Diam.  of  —  O—  circle  = 

Df 

.     With  this  Diam.  and  given  s  find  Qf  for  n  =  .015  from 

1.1056 

Diagrams  i,  2,  3  or  4.  Divide  observed  Qf  by  Qf  for  n  =  .015 
and  with  this  ratio  find  required  n  from  Diagram  5. 

In  the  last  four  cases  Af  is  obtained  from  Df  by  equations  given  on 
the  ratio  diagrams,  and  Vf  by  dividing  Qf  by  Af.  Also  since  the 
Diam.  of  —  O  —  circle  refers  to  equal  capacity  and  not  equal  area,  in 
general  Vf  cannot  be  substituted  for  Qf  as  in  the  first  four  cases. 


HYDRAULIC  DIAGRAMS.  15 

Class  B. — Conduits  flowing:  partially  full. 

Cases  arising-  in  this  class  involve  merely  an  extended  use  of  the 
ratio  diagrams,  and  the  principles  involved  above.  To  avoid  con- 
fusion the  Gothic  section  will  be  retained  as  an  example  in  the 
following1  cases  of  this  class. 

(1)  Given  Df,  Ds  and  Qf  or  Vf  or  Af  of  a  Gothic  section  to  obtain 

Q8    Vs          As 

Qs,  V*  and  As.  From  Diagram  7  obtain  — ,  —  and  —  correspond- 

Qf    Vf          Af 

Ds  Qs 

ing  to  — .     The  product  of  Qf  and  --  will  give  required  Qs ;  pro- 
Df  Qf 

Vs  As 

duct  of  Vf  and  —  required  V8 :  product  of  Af  and  --  required  A». 
Vf  Af 

(2)  Given  Df,  Ds  and  Q8  or  V»  or  A»  of  a  Gothic  section  to  obtain 

Q8   V8          A* 

Qf.  Vt  and  Af.    From,  Diagram  7  obtain  — ,  —  and  —  correspond- 

Qf   Vf          Af 

DS  Q8 

ing  to  — .     The  quotient  of  Qs  by  --  will  give  required  Qf ;  quo- 
Df  Qf 

Vs  A8 

tient  of  Vs  by  —  required  Vf ;  quotient  of  A»  by  —  required  Af. 
Vf  Af 

(3)  Given  s,  n,  D8  and  Df  of  a  Gothic  section  to  obtain  Q*.  With 
given  Df,  s  and  n  obtain  Qf  by  (A-5).     With  this  Qf  and  given  Ds 
and  Df  obtain  Qs  by  (B-i). 

(4)  Given  Q».  n,  Ds  and  Df  of  a  Gothic  section  to  obtain  s.  With 
given  Qs,  Ds  and  Df  obtain  Qf  by  (6-2).     With  Qf,  and  given  n 
and  Df  obtain  required  s  by  (A-7). 

(5)  Given  Qs  (obtained  from  direct  observation),  s,  D8  and  Df  of 
a  Gothic  section  to  obtain  n.     With  given  Df,  Ds  and  Qs  obtain  Qf 
by  (B-2).     With  this  Qf  and  given  Df  and  s  find  required  n  from 
(A-8). 

(6)  Given  Qs,  s,  n  and  Df  of  a  Gothic  section  to  obtain  D8.    With 

given  s,  n  and  Df  obtain  Qf  by  (A-s).     The  product  of  Di  and  - 

Df 

corresponding  to  —  will  give  required  Ds. 

Qf 


,6    -  HYDRAULIC   DIAGRAMS. 

(7)  Given  Qs,  s,  n  and  Ds  of  a  Gothic  section  to  obtain  Df.     A 
tentative  method    is  necessary  as  follows :  Assume   Df,  and,  with 
given  s  and  n  and  Ds,  obtain  a  trial  Q*  by  ( B-3) ,  which  should  agree 
with  given  Qs  if  Df  is  correctly  assumed  ;  if  not,  assume  new  values 
of  Df  until  the  proper  one  is  found. 

(8)  Given  —  (but  knowing  neither  D«  nor  Df),  Qs,  s  and  n  of 

Df 

Ds 
a  Gothic  section  to  obtain  Dr.  Obtain  Qf  from  Qs  and  -  -  by  (B-2), 

Df 
and  with  this  Qf  and  given  s  and  n  obtain  Df  by  (A-6). 

Class  C. — Conduits   under  Pressure. 

With  conduits  flowing  under  pressure,  the  following  losses  of 
head  are  usually  considered :  Loss  due  to  generating  velocity,  loss 
due  to  entrance,  loss  due  to  internal  friction,  and  loss  due  to  special 
causes,  such  as  valves  and  sudden  bends. 

For  uniform  flow  in  long  conduits  the  losses  due  to  entrance  and 
to  generating  velocity  are  relatively  small  and  for  entrance  similar 
to  a  standard  short  tube,  these  two  losses  combined  amount'  ap- 
V2 

proximately  to   1.5 -.       Losses  due  to  valves  and  bends,  also 

2  g 

usually  small,  and  requiring  special  treatment,  will  not  be  con- 
sidered here. 

The  principal  loss,  then,  and  the  only  one  which  is  considered 
here,  is  that  due  to  internal  friction,  and  usually  represented  by  the 
hydraulic  gradient — or  the  line  joining  water  levels  in  piezometers 
placed  at  points  along  the  conduit  and  which  for  convenience,  will 
here  be  assumed  as  straight. 

If  the  conduit  does  not  at  any  point  rise  above  this  hydraulic  gra- 
dient, and  if  we  let  s  represent  the  inclination  of  this  hydraulic  gra- 
dient, the  solution  of  cases  in  this  class  will  be  the  same  as  for  cases 
under  Class  A,  to  which  the  reader  is  referred. 

Canals. 

Under  the  present  treatment  of  canals  in  which  the  ''Filled 'Sec- 
tion" is  an  assumed  one  having  a  depth  equal  to  one-half  the  base, 
only  two  conditions  of  flow  have  a  practical  significance :  Flow  in 
which  the  ratio  of  depth  of  flow  to  depth  of  filled  section  is  T.OO  (i.  e. 


HYDRAULIC  DIAGRAMS.  17 

flowing  full),  and  flow  in  which  this  ratio  is  greater  or  less  than 
i.oo.  The  two  classes  corresponding  to  these  two  conditions  will 
be  considered  separately. 

Class  A. — Sections  in  which  the  ratio  of  depth  of  flow  to  depth 
of  "Filled  Section"  is  i.oo. 

If  the  sections  are  rectangular  the  following  four  principal  cases 
occur : 

1 i )  Given  B,  s  and  n  to  obtain  Qf.  With  given  B  and  s  obtain  Qf 
for  n  =  .025  from  Diagram  14.  The  product  of  this  Qf  and  the  ratio 
from  Diagram  15  corresponding  to  given  n  will  give  required  Qf. 

(2)  Given  Qf,  s  and  n  to  obtain  B.    Divide  Qf  by  ratio  on  Diagram 
15  corresponding  to  given  n  and  with  this  Qf,  corresponding  to  n  = 
.025,  and  given  s,  find  required   B  from   Diagram  14.     This  case 
is  also  slightly  tentative  as  in  the  case  of  conduits  (A-2)  and  an 
estimate  of  B  is  necessary  in  advance  in  order  to  use  Diagram  15. 

(3)  Given  Qf.  B,  and  n  to  obtain  s.    Divide  given  Qf  by  ratio  on 
Diagram  15  corresponding  to  given  n.     With  this  Qf  for  n  =  .025 
and  given  B  find  required  s  from  Diagram  14. 

(4)  Given  Qf  (obtained  from  direct  observation),  B  and  s  to 
obtain    n.     With   given   B    and   s    obtain   Qf  for  n  =  .025  from 
Diagram   14.     Divide  observed   Qf  by  Qf  for  n  =  .025   and  with 
this  ratio  find  required  n  from  Diagram  15. 

In  the  above  four  cases  Vf  may  be  substituted  for  Qf  for  reasons 
similar  to  those  previously  described  under  Class  A,  for  conduits, 
Af  in  all  cases  is  obtainnd  from  equations  on  Diagram  17. 

If  the  sections  are  trapezoidal  these  four  cases  are  modified,  and 
the  section  with  side  slopes  of  2  to  I  will  be  considered  an  example, 
as  follows  : 

(5)  Given  B,  s  and  n  of  a  trapezoidal  section  having  side  slopes  of 
2  to  i,  to  obtain  Qf.  From  Diagram  16  base  of  =O  =  Rect.  section 

B 

— .     With  this  base  and  given  s  find  Qf  for  n  =  .025  by  Dia- 
0.73 

gram   14.     The  product  of  this  Qf  and  the  ratio  on  Diagram  15 
corresponding  to  given  n  will  give  required  Qf. 

(6)  Given  Qf,  s  and  n  of  a  trapezoidal  section  having  side  slopes  of 
2  to  i  to  obtain  B.  With  given  Qf,  s  and  n  obtain  B  of  rectangular 
section  by  (A-2).     From  Diagram  16,  required  B  =  .73  X  Base  of 
=  C  =  Rect.  Section. 


!g  HYDRAULIC   DIAGRAMS. 

(7)  Given  Qf,  B  and  n  of  a  trapezoidal  section  with  slopes  of  2 
to  I  to  obtain  s.     From  Diagram  16  Base  of  —  O=  Rect.  Section 

B 

=  .    Also,  divide  given  Qt  by  ratio  on  Diagram  15  correspond- 

0.73 
ing  to  given  n.     With  this  Qf  for  n  —  .025  and  Base  =C=  Rect. 

Section  obtain  required  s  from  Diagram  14. 

(8)  Given  Qf  (obtained  from  direct  observation),  B  and  s  of  a 
trapezoidal  section  with  side  slopes  of  2  to  I  to  obtain  n.     From 

B 

Diagram  16,  Base  of  =  C=  Rect.  Section  = .     With  this  Base 

0-73 

and  given  s  find  Qt  for  n  ~  .025  from  Diagram  14.  Divide 
observed  Qf  by  Qf  for  n  =  .025,  and  with  this  ratio  find  required  n 
from  Diagram  15. 

In  the  last  four  cases  Vt  cannot,  in  general,  be  substituted  for  Qf 
for  reasons  similar  to  those  previously  given  under  Class  A  for 
conduits.  Af  in  all  cases  is  obtained  from  equations  on  Diagram  17, 
and  Vf  by  dividing  Qf  by  Af. 

Class  B.— Sections  in  which  the  ratio  of  depth  of  flow  to  depth  of 
"  Filled  Section"  is  greater  or  less  than  i.oo. 

1 i)  Given  Df  (or,  if  B  is  given,  Df  =  |  B),  Ds  and  Qf  or  Af  of  a 
trapezoidal  section  with  side  slopes  of  2  to  I,  to  obtain  Qs  and  As. 

Qs          As  Ds 

From  Diagrams  16  and  17  obtain  --  and  —  corresponding  to  - 

Qf          Af  Df 

Qs 

The  product  of  Qt  and  --  will  give  required  Qs ;  the  product  of  Af 
Qf 

A 

and  — -  required  As.     In  all  cases  where  Vf  and  Vs  are  sought  they 

Af 
are  obtained  by  dividing  Qf  by  Af  or  Qs  by  As,  respectively. 

(2)  Given  Df,  Ds  and  Qs  or  As  of  a  trapezoidal  section  with  side 
slopes  of  2  to  i  to  obtain  Qf  and  Af.  From  Diagrams  16  and  17  obtain 

-  and  --  corresponding  to  — .  The  quotient  of  Qs  by  -  -  will 
Qf  Af  Df  Qf 

A  • 
give  required  Qf ;  quotient  of  As  by  -  -  required  Af .    Vs  and  Vf  are 

Af 
obtained  by  division  as  before. 

(3)  Given  s,n,  Dsand  B  ( or  Df  which  equals  i  B)  of  a  trapezoidal 
section  with  side  slopes  of  2  to  i  to  obtain  Qs.    With  given  B,  s  and 


HYDRAULIC  DIAGRAMS.  19 

n  obtain  Qf  by  (A-s).     With  this  Qf  and  given  Ds  and  Df  obtain 
required  Qs  by  (B-i). 

(4)  Given  Q»,  n,  Ds  and   B  of  a  trapezoidal  section  with  side 
slopes  of  2  to  i  to  obtain  s.    From  Ds,  B  and  Qs  obtain  Qf  by  (6-2). 
With  Qf  and  given  n  and  B  obtain  required  s  by  (A-7). 

(5)  Given  Qs  (obtained  from  direct  observation),  s,  Ds  and  B  of 
a  trapezoidal  section  with  side  slopes  of  2  to  i  to  obtain  n.     With 
given  Qs,  B  and  Ds  obtain  Qf  by  (6-2).     With  this  Qf  and  given 
B  and  s  find  required  n  from  (A-8). 

(6)  Given  Qs,  s,  n  and  B   of  trapezoidal  section  with  side  slopes 
of  2  to  i  to  obtain  Ds.    With  given  s,  n  and  B  obtain  Qf  by  (A-5). 

Qs          Ds 

With  --  find  --  from  Diagram  16.     The  product  of  Df  (equal  to 
Qt          Df 

Ds 

\  B)  and  —  gives  required  Ds. 
Df 

(7)  Given  Qs,  s,  n  and  Ds  of  a  trapezoidal  section  with  side  slopes 
of  2  to  i  to  obtain  B.    A  tentative  method  is  necessary.    Assume  B, 
and,  with  given   s,  n  and   Ds,  obtain  a  trial  Qs  by  (6-3),  which 
should  agree  with  given  Qs  if  B  is  correctly  assumed  ;  if  not  assume 
new  values  of  B  until  the  proper  one  is  found. 

..  '  Ds       Ds 

(8)  Given  —  or  —  (but  knowing  neither  Ds,  Dt  nor  B),  Qs,  s 

Df        B 

and  n  of  a  trapezoidal  section  with  side  slopes  of  2  to  i  to  obtain  B. 

Ds        Ds 

With  given  Qs  and  —  or  —  obtain   Qt  by   (B-2).     With  Qf  and 
Df        B 

given  s  and  n  obtain  B  by  (A-6). 

The  following  two  numerical  examples  will  now  be  given  which 
may  further  illustrate  the  use  of  the  diagrams  of  the  two  groups. 

(Ex.  i).  Let  it  be  required  to  find  the  hydraulic  slope  of  an  egg- 
shaped  conduit  24X36  ins.,  which,  when  flowing  1.5  ft.  deep  will  dis- 
charge 4  cu.  ft.  per  sec.  with  n  =  .014.  The  case  falls  under  (6-4) 
for  conduits  in  which  Q»  =  4,  n  =  .014,  D«  =  iSins.,  Df  =  36  ins. 

D8       18  Q8 

and  required  to  obtain  s.    —  =  —  =  .500.  By  Diagram  10,  —  = 

Df       36  Qf 

0.425.     Also  Diam.   of   =C=   circle   =   •        -  =  28^  ins.     Qf  = 

1.254 


20  HYDRAULIC   DIAGRAMS. 

4.0  9.41 

=  9.41  cu.  ft.  per  sec.    Qf  (for  n  =  .015)  = =  8.6  cu.  ft. 

0.425  1.09 

per  sec.  (by  Diagram  5),  which,  with  diameter  28^  ins.,  gives  re- 
quired s  =  0.0008  =  I  in  1250  by  Diagram  3. 

(Ex.  2).  Let  it  be  required  to  find  the  discharge  of  a  trapezoidal 
canal  with  length  of  base  18  ft.,  side  slopes  I  to  i,  flowing  8  ft.  deep, 
with  s  =  i  in  5,000,  and  n  =  .020.  The  case  falls  under  (6-3)  for 
Canals.  For  the  filled  section,  then,  B  =  18  ft.,  and  D  =  9  ft. 

18 

By  Diagram  16,  Base  of  =C=  Rect.  section  =  -    -  =  22.5  ft., 

0.80 

which,  with  given  s  =  i  in  5,000,  gives  Qf  =  683  cu.  ft.  per  sec.  for 
n  =  .025  (by  Diagram  14).     Qf  (for  n  =  .020)  =  683  X  1.235  = 

D8        8 

843  cu.  ft.  per  sec.  by  Diagram  15.     —        —  =  0.889,  which,  for 

Df        9 

side  slopes  I  to  i  gives  --  =  0.81  (by  Diagram  16).     Required  Q* 

Qt 

=  843  X  0.8 1  =  683  cu.  ft.  per  sec. 

In  conclusion,  it  may  be  said  as  to  the  accuracy  of  using  these 
diagrams  that,  disregarding  personal  errors  in  reading,  there  will 
be  a  slight  disagreement  between  results  computed  by  the  general 
formula  and  those  given  by  the  diagrams,  due  to  the  nature  of  the 
formula  and  the  method  of  constructing  the  diagrams ;  that  this 
disagreement  will  be  greater  for  canals  than  for  conduits,  owing  to 
the  greater  range  of  data  considered  under  canals ;  and  that  for 
ordinary  practice  this  disagreement  should  not  be  greater  than 
one  or  two  per  cent,  for  conduits  and  two  or  three  per  cent,  for 
canals. 

Disagreements  in  this  amount  have  little  importance,  however, 
when  we  consider  that  first  the  formula  itself  is  empirical  and,  being 
based  largely  upon  experiments,  is  subject  to  necessary  errors  in 
its  deduction  ;  also,  that  the  uncertainties  in  selecting  values  for  n 
are  appreciable,  'and  are  greater  for  canals  than  for  conduits,  as 
will  be  seen  from  a  study  of  the  classified  values  of  n  in  conjunc- 
tion with  Diagrams  5  and  15. 


HYDRAULIC  DIAGRAMS. 


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E    co 


HYDRAULIC  DIAGRAMS. 


27 


I'l 

c   — 
'5    Q 


THF 

I    UNIVERSITY 

OF 

X^jjrc.Rj^ 


HYDRAULIC   DIAGRAMS. 


0.6        OS          1.0 


12          13          1.4          1.5          1.6  17          IA          1.9 


Diagram 

Showing 

Ratios  of  Discharge 

for 

Various  Friction  FactorsV 
to  Discharge  for  n-.oi5 

for 
Circular  Channels. 


as         0.9         1.0         1.1          1.2         1.3         1.4         1.5        1.6          1.7         1.8         1.9 
Ratio  of  Discharge. 
DIAGRAM  5. 


OF  THE 

UNIVERSITY 


HYDRAULIC  DIAGRAMS. 


O.I          0.2         0.3         0.4          0.5         0.6          0.7 


Hydraulic  Elements 

of 

Circular  Sections 
by  Kutter'S  formula. 


0.1          0.2         0.3         0.4- 


C.5 


Ratio  of  the  Hydraulic  Elementsof  the  F;"M  f*nm*nt  tn-those  of  the  Entire  Circle. 
DIAGRAM  6. 


0.1          02         0.3          0.4         0.5          0.6         0.7          0.8         0.9          1.0  I.I 


O.I 


O.I          0.2         0.3          0.4         0.5         0.6          0.7          0.6         0.9          1.0          II  1.2 

Ratio  of  the  Hydraulic  Elementsof  1he  filled  Segment  tolhose  of  the  Entire  Section. 
DIAGRAM  7. 


HYDRAULIC  DIAGRAMS. 


33 


0.3         0.4         0.5          0.6         0.7         0.8 


0.1          02          0.3          0.4-          0.5         0.6          0.7         0.6          0.9          1.0          \.\  1.2 

Ratio  of  the  Hydraulic  Elements  of  the  Filled  Segment  to  those  of  the  Entire  Section. 

DIAGRAM  8. 
0.1          0.2         0.3         0.4         0.5         0.6         0.7         0.8 


T  Hydraulic  Elements 
of  Catenary  Section 
byKutter's  Formula. 


-  Vert.  Diam  =0=1.063* Diam.  of  == 


0.1          0.2         0.3         0.4-         0.5         0.6         0.7         0.8         0.9          1.0          1.1  1.2 

Ratio  of  the  Hydraulic  Elements  of  the  Filled  Segment  tottiose  of  me  Entire  Section. 
DIAGRAM  9. 


HYDRAULIC  DIAGRAMS. 

O.I         0.2         0.3         0.4         0.5         0.6         0.7         0£         0.9          1.0          1.1 


35 


I.? 


O.I          O.Z          0.3          0.4          O.F         0.6          0.7          0.8         0.9          J.O  I.I  12 

Ratio  of  the  Hydraulic  Elements  of  the  Filled  Segment  to  those  of  the  Entire  Section. 

DIAGRAM  10. 
O.I  02         03          04          0.5         0.6          0.7          0.6          0.9          I.O          I.I 


0.9 

&  0.7 

$0.6 

I 
I  0-5 


0.4 


J"  Hydraulic  Elements     ;~~~1 
of  Square  Section  ] 

by  Kutrer^s  Formula.       ~--j\ 

n-.OI5  s-tsso  H'Kfo'D-IO'O'  ~—fi 
HorDiam.*H=0.9l5xDiam.of&Cinle  : 


O.I  0.^          03          0.4-         0.5          Of.          0.7          0.8          03  1.0  \\ 

Ratio  of  Hydraulic  Elements  of  the  Filled  Segment  to  those  of  theEntire  Section. 
DIAGRAM  11. 


HYDRAULIC  DIAGRAMS. 


37 


1.0 


O.I         0.2          0.3         Q.4          0.5          0£>          0.7          0.8          09  10  M          '\2 


Oft 


0.7 


Hydraulic  Elements 
of  Horseshoe  Section 


O.I  0.2         03        0.4         0.5         0.6         0.7        0.8        .0.9         1.0          I.I 

Ratio  of  Hydraulic  Elements  of  the  Filled  Segment  to  those  of  the  Entire  Section. 
DIAGRAM   12.     (WACHUSETT  AQUEDUCT.) 

O.I 


0.4         0.5         O.b         0.7         0.8         0.9          I.O          I.I 


"Hydraulic  Elements 
of  Horseshoe  Section 
by  Kutter's  Formula 
n-.oi5   5= 


Vert  Di am. *D= 0.965 *D/a.  of&Cinle 


0.1          0.2         03         OA         0.5         0.6         0.7         0.8          0.9          1.0          I.I          L2 
Ratio  of  Hydraulic  Elements  of  the  Filled  Segment  TO  Unose  of  the  Entire  Section. 
DIAGRAM   13.     (CROTON^AQUEDUCT.) 


it  1 


HYDRAULIC  DIAGRAMS. 
•sdo|g  ^ 

ru  r>J        i^  ?       S  2" 


'S    UOj4.DU.lpUl  Dj 


HYDRAULIC  DIAGRAMS. 


0.7 


0.8       0.9         1.0 


1.2        1.3         1.4        1.5 


1.6 


•u*u  35:::  ::::-:::::" 

A-XQ         JJ      f 

.0  5_ 

"tP" 

.038  -!-^\^^ 

H  r  —  IT  —  in  

.038 

.037  -    J^gf> 

--.037 

.036  T-^SrHrp 

DiacjrQm 

J.036 

.035  ii  i-^Sjj  11: 

-  Showing  Ratio  of  Discharge 

_U__-035 

for 

OTUL 

Various  Friction  Factors"n" 

~~    _  _  033 

i               j   \ 

031-        -5ev- 

to 

-  -  .032 

-  -  031 

.f 

_,,         n-tn 

A  Oft                                                           i 

-    029 

Rectangular  ana 

fi  *7fl                                    ^  i 

T                  *]P       C       j.- 

-    028 

.0  27                                              ^ 
.026--                                  --3^ 

.024  ^ 
.023  -- 
.022  '-'• 

Trapezoida  Sections. 

!  ! 

.027 
--J026 

1             .025 
--.024 
-•+-    --025 
--.022 
-  -  021 

.019  4 
.018  - 

.one: 
.016  :: 
.015  --i  

0.7        OA         0.9 

Ijlllllpijjjili 

1.0         1.1          L2         1.3         1.4         1.5         1.6 

-  -p  -  -  .020 
-  -  .019 
.018 
--.017 

r-016 

Ratio  of  Discharqe. 
DIAGRAM   15. 


HYDRAULIC  DIAGRAMS. 


43 


K>  CJ  — 


3       5 


§       S       $       8      &       5 
04  Moy  40  H4daQ  ^o  o.qjog 


HYDRAULIC  DIAGRAMS. 


45 


OF  THE 

UNIVERSITY  ] 

OF 


THIS  BOOK  IS  DUE  ON  THE  LAST  DATE 
STAMPED  BELOW 


30m-l,'15 


YC   13497 


